WASHINGTON (CNN) -- Russia's attacks against the former Soviet republic of Georgia have "substantially damaged Russia's standing in the world," President Bush told reporters Monday after returning from his trip to Asia.

President Bush tells reporters that Russia is jeopardizing its relations with the United States and Europe.

Bush also warned Russia against trying to depose Georgia's government, saying evidence suggests Russia may be preparing to do so.

"Russia must reverse the course it appears to be on," Bush said in the Rose Garden of the White House.

The Russian military advanced farther into Georgia on Monday, moving beyond Georgia's two breakaway provinces, according to Georgia's government, which has called for a cease-fire.

Russian and Georgian officials said Russia had entered Senaki, in western Georgia, outside of breakaway Abkhazia. But Russia has not confirmed any other advance of its ground forces beyond South Ossetia and Abkhazia.

"If these reports are accurate ... these actions would be inconsistent with the assurances we received from Russia that its objectives were limited to restoring the status quo in [Georgian breakaway region] South Ossetia that existed before the fighting began on August 6," Bush said. Watch Bush demand a cease-fire »

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World powers have urged Russia to agree to an immediate cease-fire with Georgia and accept international mediation on the crisis in South Ossetia. See photos from the conflict »

Bush said Russia's invasion of Georgia and the threat to Georgia's democratically elected government "is unacceptable in the 21st century" and has "substantially damaged Russia's standing in the world."

"These actions jeopardize Russians' relations ... with the United States and Europe," Bush said.

Earlier Monday, Russian Prime Minister Vladimir Putin said his nation will take its efforts to their "logical conclusion."

Russian officials insisted their efforts were aimed at stopping Georgian military actions against Russian peacekeepers in the breakaway regions and against Ossetians loyal to Russia. Over several years, Russia has given passports to many residents of South Ossetia -- which borders Russia -- and declared them Russian citizens.

Georgia's ambassador to Washington said he hopes Putin "will listen to what President Bush had to say."

"The time is very critical for my country," Vasil Sikharulidze said. "Georgia faces extinction under this brutal Russian invasion."

The conflict began last week after Georgia said it launched a military operation Thursday in South Ossetia after artillery fire from separatists killed 10 people.

Russia sent its forces there Friday, accusing Georgia of plotting to wipe out Ossetians loyal to Russia. By the weekend, Russia had sent forces to Abkhazia, a second breakaway Georgian territory.

Georgia has contended that Russia merely wants control of an oil pipeline, and Georgian President Mikheil Saakashvili said Monday that Russia seeks to stamp out his country's "democratic ambitions."

South Ossetia's capital, Tskhinvali, lay in smoldering ruins Monday after four days of fighting. Each side accused the other of killing large numbers of civilians. Russia said at least 2,000 people had been killed in Tskhinvali.

At the United Nations in New York on Monday, Russia refused to sign off on a resolution calling for a cease-fire with Georgia despite mounting pressure from world powers.

Russian Ambassador Vitaly Churkin said the proposed U.N. resolution, drafted by French officials, was lacking in a "serious number" of areas.

"I can't see us accepting this French draft of this resolution," Churkin told reporters late Monday. "We will look at the draft and try to bring it to a standard where it can play a role in this."

One of the issues Churkin mentioned was that the draft resolution, which has not been made public, did not mention Georgia's previous "aggression" in South Ossetia. Russia argues that it is acting in response to a Georgian attack on the territory, whose separatist government is backed by Moscow.

The details of the draft -- backed by the United States and the other European members of the Security Council -- include calls for an immediate cease-fire, a complete withdrawal of Russian and Georgian forces and participation in mediation. The council could take up the proposal on Tuesday -- but as one of five permanent members of the Security Council, Russia could veto any resolution.

U.S. Secretary of State Condoleezza Rice spoke by telephone Monday with most of the foreign ministers belonging to the Group of Eight leading industrialized nations about the conflict.

The G-8 consists of the U.S., Britain, France, Germany, Italy, Canada, Japan and Russia. Russia, which officially joined the group in 1998, was absent from Monday's discussions.

The ministers reiterated their support for Georgian sovereignty and called for a diplomatic solution, Deputy State Department spokesman Robert Wood told reporters.

Rice and the ministers voiced concern over the increasing civilian casualties and pressed Russia to accept a cease-fire.

Saakashvili said Monday he signed an internationally brokered cease-fire proposal that was to be taken to Moscow on Monday evening by French Foreign Minister Bernard Kouchner and Finnish Foreign Minister Alexander Stubb.

NATO also is involved in efforts to broker a deal between Georgia and Russia.

The U.S. is trying to transport 2,000 Georgian troops serving in Iraq back home to fend off Russian advances, but beyond that "there are no discussions about the U.S. getting involved militarily," Wood said.

The United States has begun to provide humanitarian assistance to Georgia, however. The first package of supplies is expected to be depleted by Monday evening.

The U.S. Embassy in Tbilisi declared a "disaster" in Georgia and has authorized $250,000 in initial funding for supplies to assist about 10,000 people.

Wood said the U.S. is pre-positioning other supplies such as tents and blankets and bringing in additional replacement supplies from Germany.

Over the weekend, the State Department allowed families of U.S. diplomats in Georgia to leave the country. Diplomats, however, would remain, the department said.

The United States also has helped about 170 Americans leave Georgia in two convoys, with a third planned Tuesday.